Subgroups — #31782573

Order 17536.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577441	C₂	1	2	yes	maximal
2	#22577231	C₂	1	2	yes	maximal, complemented
3	#22577229	C₂	1	2	yes	maximal, complemented
4	C₂²×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#17984047	—	2	2	no	—
6	#17984022	C₂²	1	4	yes	—
7	#17984020	—	2	2	no	—
8	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₁₃₇	C₄	1	4	yes	complemented
10	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₃₇	—	2	4	no	—
13	C₂²⋊Dic₁₃₇	—	4	2	no	—
14	C₄×Dic₁₃₇	—	4	2	no	—
15	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₃₇	Q₈	1	2	yes	—
19	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
20	C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
21	C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
22	C₂×Dic₁₃₇	—	4	4	no	—
23	C₂³×C₁₃₇	D₈	1	1	yes	characteristic
24	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
26	C₂×C₄×C₁₃₇	Q₁₆	1	2	yes	—
27	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
28	C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
29	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
30	C₈×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₁₃₇	—	2	2	no	—
32	C₂²×C₁₃₇	#107	1	2	yes	—
33	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
34	C₄×C₁₃₇	#107	1	2	yes	—
35	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
37	Dic₁₃₇	—	8	4	no	—
38	C₂×C₁₃₇	—	2	2	no	—
39	C₂×C₁₃₇	C₂×#107	1	1	yes	characteristic
40	C₂×C₁₃₇	#508	1	2	yes	—
41	C₁₃₇	#2158	1	1	yes	characteristic, complemented, Sylow
42	#2158	—	137	1	no	maximal, Sylow
43	C₂²×C₁₆	D₁₃₇	1	1	yes	characteristic
44	#427	—	137	2	no	—
45	#367	—	137	2	no	—
46	#366	—	137	2	no	—
47	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
48	#117	—	274	2	no	—
49	C₂×C₁₆	Dic₁₃₇	1	4	yes	complemented
50	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
51	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
52	#107	—	137	4	no	—
53	#106	—	274	2	no	—
54	C₂²×C₄	D₅₄₈	1	1	yes	characteristic
55	C₂×C₈	C₂×Dic₁₃₇	1	2	yes	—
56	C₂×C₈	D₅₄₈	1	1	yes	characteristic
57	C₂×C₈	Dic₂₇₄	1	2	yes	—
58	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
59	C₄⋊C₄	—	274	4	no	—
60	C₂²⋊C₄	—	548	2	no	—
61	C₄²	—	548	2	no	—
62	C₁₆	C₂×Dic₁₃₇	1	2	yes	—
63	C₁₆	C₂×Dic₁₃₇	1	2	yes	—
64	C₂³	D₁₀₉₆	1	1	yes	characteristic
65	C₂×C₄	—	548	4	no	—
66	C₂×C₄	C₄⋊Dic₁₃₇	1	2	yes	—
67	C₂×C₄	D₁₀₉₆	1	1	yes	characteristic
68	C₂×C₄	Dic₅₄₈	1	2	yes	—
69	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
70	C₈	C₄⋊Dic₁₃₇	1	2	yes	—
71	C₈	C₂²×Dic₁₃₇	1	1	yes	characteristic
72	C₈	C₂×Dic₂₇₄	1	1	yes	characteristic
73	C₂²	—	2	2	no	—
74	C₂²	#17984022	1	2	yes	—
75	C₂²	C₂×D₁₀₉₆	1	1	yes	characteristic
76	C₄	—	1096	4	no	—
77	C₄	#17984022	1	2	yes	—
78	C₄	C₂×C₄⋊Dic₁₃₇	1	1	yes	characteristic
79	C₄	C₂×Dic₅₄₈	1	1	yes	characteristic
80	C₂	—	2	2	no	—
81	C₂	C₂×#17984022	1	1	yes	characteristic
82	C₂	#22577835	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₁₀₉₆
commutator	C₈×C₁₃₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	C₂²×C₁₆×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₈
prime radical	C₂³×C₁₃₇	D₈
cosocle	C₂×C₈×C₁₃₇	C₂³
hypercenter	C₂²×C₁₆	D₁₃₇
hypocenter	C₁₃₇	#2158
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #22577231	C₂	semidirect
3. #22577229	C₂	semidirect
9. C₂×C₁₆×C₁₃₇	C₄	semidirect
41. C₁₃₇	#2158	semidirect
49. C₂×C₁₆	Dic₁₃₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22577441	2
#22577231	2
#22577229	2
C₂²×C₁₆×C₁₃₇	2
#2158	137